## A quasi-spectral characterization of strongly distance-regular graphs.(English)Zbl 0956.05103

Electron. J. Comb. 7, No. 1, Research paper R51, 9 p. (2000); printed version J. Comb. 7, No. 2 (2000).
A graph $$\Gamma$$ with diameter $$d$$ is strongly distance-regular if $$\Gamma$$ is distance-regular and its distance-$$d$$ graph is strongly regular. It has been conjectured that a strongly distance-regular graph is antipodal or has diameter at most three. Let $$\Gamma$$ be a graph with spectrum $$\{\lambda_0^{m_0},\dots,\lambda_d^{m_d}\}$$, where $$\lambda_0>\cdots >\lambda_d$$, and let $$n=|V(\Gamma)|$$. Let $$\pi_i=\prod_{j\neq i}|\lambda_i-\lambda_j|$$, $$\sigma_e=m_2+m_4+\cdots$$, $$\sigma_o=m_1+m_3+\cdots$$, $$\Sigma_e=\pi_0/\pi_2+\pi_0/\pi_4+\cdots$$, $$\Sigma_o=\pi_0/\pi_1+\pi_0/\pi_3+\cdots$$, for $$u\in V$$ let $$k_{d-1}(u)= |\Gamma_{d-1}(u)|$$ and let $$H={\displaystyle \frac{n}{\sum_{u\in V}1/k_{d-1}(u)}}$$. The main result of this paper is the following Theorem 2.2: A regular graph $$\Gamma$$ with $$n$$ vertices, eigenvalues $$\lambda_0>\cdots >\lambda_d$$, and parameters $$\sigma_e$$ and $$\Sigma_e$$ as above, is strongly distance-regular if and only if $H=n-\frac{n\sigma_e \sigma_o}{n\Sigma_e \Sigma_o+(\sigma_e-\Sigma_e) (\sigma_o+\Sigma_o)}.$

### MSC:

 05E30 Association schemes, strongly regular graphs 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C75 Structural characterization of families of graphs
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